Study of Caterpillar-like Motion of a Four-link · PDF fileStudy of Caterpillar-like Motion of a Four-link Robot . S.F. Jatsun, L.Yu. Vorochaeva, S.I. Savin, A.S. Yatsun . ... employed

The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS13.070

Study of Caterpillar-like Motion of a Four-link Robot S.F. Jatsun, L.Yu. Vorochaeva, S.I. Savin, A.S. Yatsun

Department of Mechanics, Mechatronics and Robotics, South-West State University, Kursk, Russia

e-mail: [email protected]

Abstract: In this paper we study a four link robot that performs

caterpillar-like motion. The device moves on a rough horizontal

surface due to friction forces that are applied to the robot at the

points of contact. The specific feature of the robot is that it has

active supports that allow control over coefficients of dry friction.

The object’s mathematical model is developed, the stages of

motion singled out, an algorithm for realizing caterpillar-like

motion and the results of numerical modeling are presented.

Keywords: caterpillar-like robot, motion stages, active and

passive supports, control torques.

1 Introduction The design of bionic robots whose motion is based on

animals is one of the important areas of development in

modern mechatronics and robotics. Caterpillar-like robots

fall into one of the broad classes of such robots.

Multilink mechanisms that describe the more

important aspects of robots of this type are used as

mathematical approximations i.e. models. These

approximations allow us to study the most important

features of the robot’s dynamics. Each link of the

multilink mechanism is presented as an absolutely rigid

body with finite mass. This allows the links that form the

mechanism to rotate relative to each other, implementing

different kinds of gaits [1-13].

Paper [1] proposes a joint torque control method based

on the assumption that there is only one active joint in the

four-link mechanism executing the climbing gait. Besides

the active joint the other three joints are all considered as

passive joints whose torques tend to zero, although they

are driven by motors in reality. Article [13] presents the

application of developing and employing modular robots

for the research of caterpillar-like motion. First an

investigation on the locomotion kinematics adopted by

natural caterpillars is given systematically. Paper [4]

describes some of the biomechanics of caterpillar

locomotion and gripping. It then describes recent work to

build a multifunctional robotic climbing machine based on

the biomechanics and neural control system

(neuromechanics) of caterpillars, Manduca sexta. In [5] a

rope climbing robotic caterpillar was designed and

achieved by imitating the gait of a natural caterpillar. A

simple scalable sinusoidal oscillator is successfully

employed for implementing diverse bionic locomotion

patterns including caterpillar-like, millipede-like, and

earthworm-like motions as described in [6].

This paper is dedicated to studying the motion of four–

link caterpillar-like robot equipped with devices that

enable it to change the way it interacts with the supporting

surface.

2 Description of the caterpillar-like robot

A diagram of the caterpillar-like robot is presented on the

figure 1. The number of the device’s links is chosen equal

to four: the extreme among them are the “head” and “tail”,

while the middle ones form the "складывающуюся

section", that provides transverse pull-up of the “tail” to

the “head” and straightening of the links into one line.

This is the minimum possible number of links required to

execute such caterpillar-like motion during which the

“head” and “tail” can exchange places as a result of which

the object moves backwards and forwards. The object can

move on an absolutely solid rough surface with no elastic-

dissipative properties, for example on asphalt, concrete

and ice. Bodies 1 and 2, 2 and 3, 3 and 4 are connected to

each other via rotational motors 5, 6 and 7. The interaction

of the robot with the supporting surface occurs at four

points via supports 8-11 which are mounted on links 1 and

4. The difference between this robot and other known

designs is the possibility to control friction acting at

supports 9 and 10. This is possible by the use of special

motors that can change the properties of the contact

surfaces of the supports. [14, 15].

Figure 1 Diagram of the robot

Let’s consider the design of the supports. Supports 9

and 10 (figure 2) consist of frame 1 which is rigidly

connected to the lower part, 2 of the corresponding unit of

the four-link robot, springs 6 and 7, electromagnetic motor

3 and metal armature 4 with sharp tip point 5 mounted on

it.

Figure 2 Central support

When the coils of the electromagnets are powered a

magnetic flux directed perpendicular to the supporting

surface is induced which causes the metal armature move

downwards, so the points clings to the supporting surface

thereby considerably increasing the coefficient of dry

friction. If the design of the support meets certain

conditions it is possible to completely fixate the central

body onto the surface. This in particular depends on the

choice of the material of the point’s tip.

Figure 3 Passive support

Passive supports 8 and 11 (figure 3) consists of frame

1 which is rigidly connected to lower part 2 of the

corresponding link and spherical joint 3 that provides a

low coefficient of friction when the support is sliding

along the surface.

Thus it is possible to change friction coefficients at

supports 9 and 10 at different stages of the motion

depending on the chosen control method.

3 Mathematical model of the robot

The robot moves in such a way that one of its extreme

links is periodically fixed onto the surface by means of

special adhesion systems. Thus the robot at every point in

time is a three-link mechanism moving relative to the

currently fixed link. To describe the motion of the four-

link robot we introduce a stationary absolute coordinate

system, Oxy and relative coordinate systems, Oixiyi which

are rigidly linked with points О1, О2, О3 and О4 in such a

way that the Oixi axis is directed along the corresponding

links (as shown in figure 4). Angles φi describe the

rotations of the Oixiyi coordinate systems relative to Oxy.

Figure 4 Analytical diagram of the mechanism

We will assume that all of the robot’s links are

absolutely rigid bodies and rods of li and whose mass mi is

concentrated at their centers of symmetry, Сi (i=1-4).

Motors 5, 6 and 7 are located at the points О2, О3 and О4

and generate torques: М12(М21), М23(М32) and М34(М43).

Active supports 9 and 10 are also located at these same

points while passive supports 8 and 11 - at points О1 and

О5.

To write a generalized mathematical model of the

robot’s motion in the vertical plane, Оху we look at the

case when link 1 is fixed onto the horizontal surface and

that the coordinates of its center of mass and angle of

inclination to the horizontal plane are constant:

constxC 1 , constyC 1 and const1 . Stationary

points О1 and О2 in fig. 4 are fixed. Taking into

consideration the constraints imposed on the system the

generalized coordinates are the links’ angles of rotation, φi,

i=2-4:

T432 q (1)

Differential equations of the system’s motion obtained

using second order Lagrange equations have the following

form:

224223

2223212

42424

2

4

3232432

3

424244

3232433

2

24322

coscos

2/cos

2/)sin(

)sin()2(

2/)cos(

)cos()2/(

)(

glmglm

glmMM

llm

llmm

llm

llmm

lmmJ

(2)

3343334323

43432

2

2

233243

2

2

434344

2332432

2

34

2

3333

cos2/cos

2/)sin(

2/)sin()2(

2/)cos(

2/)cos()2(

)4/(

glmglmMM

llm

llmm

llm

llmm

lmlmJC

(3)

2/cos2/)sin(

2/)sin(2/)cos(

2/)cos(4/

4443434434

2

3

24424

2

2344343

244242

2

4444

glmMllm

llmllm

llmlmJC

(4)

where 3CJ , 4CJ are the central moments of inertia of the

links, 2J - the moment of inertia of link 2 relative to point

О2.

4 Stages of motion for caterpillar-like motion of the

robot

To implement caterpillar-like motion we will use the

following cyclogram for control inputs (see figure 5).

Figure 5 Cyclogram for control torques M12, M43 and

electromagnetic forces F2 and F4 at contact elements during

caterpillar-like motion

The principle of this type of motion lies in periodic

relative motion of one of the robot’s links with respect to

the others under the action of torques M12 and M43

generated by motors located at points О2 and О4 and also

in the successive connection-disconnection of contact

elements that enable friction control. In this work we

assume that fixation of the contact elements onto the

surface is implemented when they are connected ( af )

and that motion on a smooth surface ( 0pf ) is

executed when they interact with the surface, where af

and pf are coefficients of friction at active and passive

supports respectively.

There are two conditions that should be met before we

can fixate links 1 and 4 by means of frictional forces

controlled by electromagnets: links 2 and 3 should be

oriented in such way that their angles of rotation vary in

certain intervals (as shown by expressions (5) and (6));

links 2 and 3 should rotate in specified directions. Thus

kinematic constraints are periodically imposed on the

mechanical system which corresponds to the fact that the

velocities of points О2, О4 are equal to zero under the

action of forces F2 and F4:

0

0

0

,0

0,

222

222

22

22

0

2

2

kn

kn

kn

kn

if

ifF

F (5)

0

0

0

,0

0,

333

333

33

33

0

4

4

kn

kn

kn

kn

if

ifF

F (6)

where φ2n, φ3n are the initial values of the angles of

rotation of links 2 and 3, φ2k, φ3k -the final values of these

angles, F20, F4

0 – certain fixed values of friction.

Torques M12 and M43 can be calculated using the

following expressions:

0

0

0

,0

0,

222

222

22

22

0

12

12

kn

kn

kn

kn

if

ifM

M (7)

0

0

0

,0

0,

333

333

33

33

0

43

43

kn

kn

kn

kn

if

ifM

M (8)

where 0

12M , 0

43M are some constant values of torques.

The caterpillar-like motion can be represented as a

sequence of stages (see figures 6 and 7).

a

b

c

Figure 6 Sequence of the robot’s positions during the first

stage of motion: a) initial position; b) intermediate position;

c) final position

a

b

c

Figure 7 Sequence of the robot’s positions during the second

stage of motion: a) initial position; b) intermediate position,;

c) final position

In figures 6 and 7 the black triangles represent points

that are assumed to be fixated.

The four link mechanism is initially at rest with its

entire links parallel to the Ox axis. In this state we have

φ1=φ2=φ3=φ4.

Active support 10 that controls friction force at point

О4 is switched on during the first stage of motion and so

friction coefficient at this point has its maximum value.

Active support 9 is switched off and link 1 is in contact

with the supporting surface via passive support 8. Torque

М43 that forces link 3 to rotate around point О4 through

angle φ3=φ3k, where φ3k is the required value of this angle,

begins to act at point О4. Passive support 8 which is

mounted at point О1 slides along the surface and link 1

remains parallel to the Ox axis (see figure 8).

Figure 8 Schematic representation of the caterpillar-like

robot during the first stage of motion

During this stage the position of the mechanism can be

described using one generalized coordinate, 2 which can

be determined by one differential equation:

2/coscoscos

2/coscos

2/cos

2/)sin(

2/)cos()(

1122331

22332

33343

23322

2

2

323222

2

3233

lllgm

llgm

glmM

llm

llmlmJ

(9)

where J3 is the moment of inertia of the link 3 with respect

to point О3.

In the second stage connected and disconnected active

supports exchange places and now point 2 is fixated onto

the surface. Torque 43M ceases to act, while torque 12M

starts acting at point О2. It is necessary to point out that

the torques are switched on and off simultaneously and

that they are equal in magnitude and direction. In this

stage link 2 executes angular motion relative point О2

until the following condition holds: k22 , where k2

is the required value. Links 2, 3 and 4 form a crank-slide

mechanism, where link 2 is the crank, link 3 – the

connecting rod and link 4 –the slider. Link 4 interacts with

the surface at point О5 by means of a passive support. As a

result of this stage the robot’s links are aligned along the

Ох axis, occupying a position similar to the initial one (fig.

9).

Figure 9 Schematic representation of the caterpillar-like

robot during the second stage of motion

In this stage we calculate the generalized coordinate,

φ2 using the following differential equation:

2/coscoscos

2/coscos

2/cos

2/)sin(

2/)cos(

4433224

33223

22212

32323

2

3

323233

2

2322

lllgm

llgm

glmM

llm

llmlmJ

(10)

where J2 is the moment of inertia of the link 2 with

respect to point О2.

5 Numerical simulation of the robot’s caterpillar-like

motion

The algorithm for simulating the robot’s caterpillar-like

motion is shown in the figure 10. The simulation uses

iterative algorithm and at time t0=0 s the mechanism is in

its initial state. To determine the stage of the motion we

use counter n. When n=1 we have the first stage of motion

and when n=2 - the second. At each moment in time the

characteristics of the system are calculated by formulas

corresponding either to the first or second stage. In the

first stage the counter’s value is determined based on the

orientation of the link 3 i.e. angle 3

. As soon as n=2 the

motion progresses to the second stage where the counter

can assume values n=1 and n=2 depending on the

orientation of the link 2 which is determined by angle φ2.

Figure 10 Algorithm used to simulate robot’s movement

Masses of the links are given by expression mi=0.1 kg

and the lengths of the links - l1=l4=0.1 m, l2=l3=0.3 m. the

initial values for the generalized coordinates and their

derivatives are equal to zero. The required values of

angles: 18/732 kk radians and torques

generated by electric motor: 8,10

43

0

12 MM Nm. The

numerical solutions of the differential equations are

obtained using Mathcad software by implementing a

numerical integration algorithm that assumes the

generalized accelerations to be constant in intervals [ti,

ti+h], where h is a constant time step.

The simulation results are presented in the form of

time graphs of angular and linear displacements (see

figures 11-13).

In figure 11 we can see that the angular displacements

of links 2 and 3 change in antiphase by the same laws and

links 1 and 4 move parallel to the surface, so 041

rad.

Figure 11 Time graphs of the links’ angular

displacements 1 – φ1(t), 2 – φ2(t), 3 – φ3(t), 4 – φ4(t)

From the graphs in fig. 12 it can be seen that during

the robot’s motion links 1 and 4 are fixated to the surface

in turns. The centers of mass of links 2 and 3 move along

the horizontal axis on trajectories with the same form, but

shifted relative to each other by time equal to the duration

of one stage of the motion.

Figure 12 Time graphs of the x-coordinates of the

centers of mass of links 1-4: 1 – xC1(t), 2 – xC2(t), 3 – xC3(t), 4 –

xC4(t)

The graphs of the ordinates of the centers of mass of

links 2 and 3 are the same. Links 1 and 4 are not separated

from the surface: yC1=yC4=0 m (see figure 13).

Figure 13 Time graphs of the y-coordinates of centers of

mass of links 1-4: 1 – yC1(t), 2 – yC2(t), 3 – yC3(t), 4 – yC4(t)

The motion of the links’ extreme points along the Ох

axis is shown in figure 14.

Figure 14 Time graphs of the x-coordinates of the extreme

points of links 1-4: 1 – xО1(t), 2 – xО2(t), 3 – xО3(t), 4 –

xО4(t), 5 – xО5(t)

Point О3 at which links 2 and 3 are connected to each

other never ceases to move along the horizontal axis.

Besides this we observe its ascent along the Оу axis in the

first stage of motion and its descent in the second stage

until it touches the surface.

Points О1 and О2 move in an identical way and the

same goes for points О4 and О5. The graphs do not

overlay each other because of the different initial positions

of these points. In figure 14 we can see repeating pattern

of fixation states and motion of the points: points О1 and

О2 move during the first stage and points О4 and О5 -

during the second stage. Motion of the points mentioned

above in the vertical direction is absent as shown in figure

15.

Figure 15 Time graphs of the y-coordinates the extreme

points of links 1-4: 1 – уО1(t), 2 – уО2(t), 3 – уО3(t),

4 – уО4(t), 5 – уО5(t)

Figures 16 and 17 show the graph of the average

velocity of the robot as a function of control torques and

the value of angle k2 .

Figure 16 Graph of ),( 043

012 MMvsr

Average speed of the robot increases by a curvilinear

law as the magnitudes of torques 4312,MM increase. The

convexity of this curve is directed downwards. From 2

Nm onwards the graph becomes linear.

Figure 17 Graph of )( 2ksrv

The graph of the average velocity as a function of

angle φ2k is also curvilinear, but as the angle increases the

average speed falls at first, attains a minimum at φ2k=550

and then rises. For values greater than φ2k=550 average

speed growth with growing φ2k. Based on the analysis of

figure 17 we can conclude that in order to achieve

caterpillar-like motion with the highest average speed the

value of angle φ2k has to be minimal.

6 Conclusions

In this paper mathematical model that describes

motion of a four link mechanism was presented. The

mechanism interacts with the surface by means of passive

and active supports and the latter allow the robot to

control friction coefficients. The design of these contact

elements and an algorithm that implements caterpillar-like

motion was presented. The results of numerical simulation

were shown in the form of time graphs of linear an

angular displacements and graphs of the robot’s average

speed as a function of variable parameters.

Work is performed by Russian Science Foundation,

Project № 14-39-00008.

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